The infinitary \(n\)-cube shuffle (Q2218659)

Higher homotopy groups of a pointed topological space \((X,x_0)\) are commutative, a fact that can be proved by rearranging finitely many cubes on the domain of a map \((S^n, s_0) \to (X,x_0)\). In spaces which are not locally contractible, there is often also a topologically significant operation of concatenation of infinitely many maps \((S^n, s_0) \to (X,x_0)\), as long as the maps grow ``ever smaller'' in a certain way. A natural question that arises in this setting is whether it would be possible to reshuffle such a concatenation in a reasonable way. This amounts to a rearrangement of infinitely many cubes on the domain of a map \((S^n, s_0) \to (X,x_0)\). Such a rearrangement has been constructed in [\textit{K. Eda} and \textit{K. Kawamura}, Fundam. Math. 165, No. 1, 17--28 (2000; Zbl 0959.55010)] and generalized in [\textit{K. Kawamura}, Colloq. Math. 96, No. 1, 27--39 (2003; Zbl 1037.55003)]. In this paper the author provides a simpler and more general proof, with the generality referring to the fact that a reshuffle remains constant on a predetermined set of finitely many cubes whose complement is path connected. The author concludes by relating his results the higher loop space structure.